## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Dale' and 'hours worked by Paul', which we can denote as $x_1$ and $x_2$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to minimize is $6x_1x_2 + 2x_2$.

## Step 3: List the constraints
The constraints given are:
- $3x_1 \leq 138$
- $12x_1 \leq 152$
- $17x_2 \leq 138$
- $6x_2 \leq 152$
- $3x_1 + 17x_2 \geq 45$
- $3x_1 + 17x_2 \geq 45$ (redundant with the previous one)
- $12x_1^2 + 6x_2^2 \geq 59$
- $12x_1 + 6x_2 \geq 59$
- $-4x_1^2 + 3x_2^2 \geq 0$
- $3x_1 + 17x_2 \leq 79$
- $12x_1 + 6x_2 \leq 146$
- $x_1$ is an integer
- $x_2$ can be a fraction

## 4: Correct and simplify the constraints
Correcting and simplifying:
- The productivity rating constraints: $3x_1 \leq 138$ and $17x_2 \leq 138$
- The organization score constraints: $12x_1 \leq 152$ and $6x_2 \leq 152$
- Combined productivity rating: $3x_1 + 17x_2 \geq 45$ and $3x_1 + 17x_2 \leq 79$
- Combined organization score: $12x_1 + 6x_2 \geq 59$ and $12x_1 + 6x_2 \leq 146$
- Other constraints: $12x_1^2 + 6x_2^2 \geq 59$, $-4x_1^2 + 3x_2^2 \geq 0$

## 5: Write down the symbolic representation
```json
{
    'sym_variables': [('x1', 'hours worked by Dale'), ('x2', 'hours worked by Paul')],
    'objective_function': '6*x1*x2 + 2*x2',
    'constraints': [
        '3*x1 <= 138',
        '12*x1 <= 152',
        '17*x2 <= 138',
        '6*x2 <= 152',
        '3*x1 + 17*x2 >= 45',
        '3*x1 + 17*x2 <= 79',
        '12*x1 + 6*x2 >= 59',
        '12*x1 + 6*x2 <= 146',
        '12*x1^2 + 6*x2^2 >= 59',
        '-4*x1^2 + 3*x2^2 >= 0',
        'x1 % 1 == 0'  # x1 is an integer
    ]
}
```

## 6: Implement the problem in Gurobi
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()
    
    # Define variables
    x1 = model.addVar(name="x1", vtype=gurobi.GRB.INTEGER)  # hours worked by Dale
    x2 = model.addVar(name="x2")  # hours worked by Paul
    
    # Objective function
    model.setObjective(6 * x1 * x2 + 2 * x2, gurobi.GRB.MINIMIZE)
    
    # Constraints
    model.addConstr(3 * x1 <= 138)
    model.addConstr(12 * x1 <= 152)
    model.addConstr(17 * x2 <= 138)
    model.addConstr(6 * x2 <= 152)
    model.addConstr(3 * x1 + 17 * x2 >= 45)
    model.addConstr(3 * x1 + 17 * x2 <= 79)
    model.addConstr(12 * x1 + 6 * x2 >= 59)
    model.addConstr(12 * x1 + 6 * x2 <= 146)
    model.addConstr(12 * x1**2 + 6 * x2**2 >= 59)
    model.addConstr(-4 * x1**2 + 3 * x2**2 >= 0)
    
    # Solve the model
    model.optimize()
    
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Dale: {x1.varValue}")
        print(f"Hours worked by Paul: {x2.varValue}")
        print(f"Objective function value: {model.objVal}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("The model is infeasible.")
    else:
        print("The model has a non-optimal status.")

solve_optimization_problem()
```